Fourth Painlev\'e Equation and $PT$-Symmetric Hamiltonians

• URL: http://arxiv.org/abs/2107.04935v2
• Date: Tue, 3 Aug 2021 21:16:09 GMT
• Title: Fourth Painlev\'e Equation and $PT$-Symmetric Hamiltonians
• Authors: Carl M. Bender and J. Komijani
• Abstract summary: This paper is an addendum to earlier papers citeR1,R2 in which it was shown that the unstable separatrix solutions for Painlev'e I and II are determined by $PT$-symmetric Hamiltonians. The constants $B_rm IV$ and $C_rm IV$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painlev'e IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=frac12
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper is an addendum to earlier papers \cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painlev\'e I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painlev\'e transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y'(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y'(0)=0$, a discrete set of initial values $y(0)=c_n$ give rise to separatrix solutions. For Painlev\'e IV the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\rm IV}n^{3/4}$ and that of $c_n$ is $c_n\sim C_{\rm IV} n^{1/2}$. The constants $B_{\rm IV}$ and $C_{\rm IV}$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painlev\'e IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=\frac{1}{2} p^2+\frac{1}{8} x^6$.

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